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Composites: Part B 42 (2011) 1276–1284
Contents lists available at ScienceDirect
Composites: Part B
journal homepage: www.elsevier.com/locate/compositesb
Analysis of laminated shells by a sinusoidal shear deformation theory
and radial basis functions collocation, accounting for through-the-thickness
deformations
A.J.M. Ferreira a,⇑, E. Carrera c, M. Cinefra c, C.M.C. Roque b, O. Polit d
a
Departamento de Engenharia Mecânica, Faculdade de Engenharia da, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal
INEGI, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal
Departament of Aeronautics and Aerospace Engineering, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy
d
Université Paris Ouest – Nanterre, 50 rue de Sevres, 92410 Ville d’Avray, France
b
c
a r t i c l e
i n f o
Article history:
Received 26 October 2010
Received in revised form 23 January 2011
Accepted 27 January 2011
Available online 26 February 2011
Keywords:
A. Laminates
B. Vibration
C. Numerical analysis
a b s t r a c t
In this paper, the static and free vibration analysis of laminated shells is performed by radial basis functions collocation, according to a sinusoidal shear deformation theory (SSDT). The SSDT theory accounts
for through-the-thickness deformation, by considering a sinusoidal evolution of all displacements with
the thickness coordinate. The equations of motion and the boundary conditions are obtained by the Carrera’s Unified Formulation, and further interpolated by collocation with radial basis functions.
Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction
The efficient load-carrying capabilities of shell structures make
them very useful in a variety of engineering applications. The continuous development of new structural materials leads to ever
increasingly complex structural designs that require careful analysis. Although analytical techniques are very important, the use of
numerical methods to solve shell mathematical models of complex
structures has become an essential ingredient in the design
process.
The most common mathematical models used to describe shell
structures may be classified in two classes according to different
physical assumptions: the Koiter model [1], based on the Kirchhoff
hypothesis and the Naghdi model [2], based on the Reissner–Mindlin assumptions that take into account the transverse shear
deformation. In this paper, the Unified Formulation (UF) by Carrera
[3–8] is proposed to derive the equations of motion and boundary
conditions. This formulation contains a large variety of 2D models
that differ in the order of used expansion in thickness direction and
in the manner the variables are modelled along the thickness. In
particular, the UF is here applied to analyse laminated shells by radial basis functions (RBF) collocation, according to a sinus-based
shear deformation theory that accounts for through-the-thickness
⇑ Corresponding author.
E-mail address: ferreira@fe.up.pt (A.J.M. Ferreira).
1359-8368/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compositesb.2011.01.031
deformations. This theory is an expansion of the developments by
Touratier [9–11], and Vidal and Polit [12]. It allows a sinusoidal
variation of all displacement components along the thickness and
it is more convenient than the classical Taylor polynomials because
the sine function can be expressed by means of Taylor expansion.
Moreover, the derivative of displacements in the deformations
does not reduce the order of the approximating functions. The
choice of the sine function can be justified from the three-dimensional point of view, using the work of Cheng [13]. As it can be seen
in Polit [14], a sine term appears in the solution of the shear equation (see Eq. (7) in [2]). Therefore, the kinematics proposed can be
seen as an approximation of the exact three-dimensional solution.
Furthermore, the sine function has an infinite radius of convergence and its Taylor expansion includes not only the third-order
terms but all the odd terms.
The most common numerical procedure for the analysis of the
shells is the finite element method [15–19]. It is known that the
phenomenon of numerical locking may arise from hidden constrains that are not well represented in the finite element approximation and, in scientific literature, it is possible to find many
methods to overcome this problem [20–25]. The present paper,
that performs the bending and free vibration analysis of laminated
shells by collocation with radial basis functions, avoids the locking
phenomenon. A radial basis function, /(kx xjk) is a spline that depends on the Euclidian distance between distinct data centers
xj;j ¼ 1; 2; . . . ; N 2 Rn , also called nodal or collocation points. The
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so-called unsymmetrical Kansa method, introduced by Kansa [26],
is employed. The use of radial basis function for the analysis of
structures and materials has been previously studied by numerous
authors [27–41]. The authors have recently applied the RBF collocation to the static deformations of composite beams and plates
[42–44].
In this paper it is investigated for the first time how the Unified
Formulation can be combined with radial basis functions to the
analysis of thin and thick laminated shells, using the SSDT, allowing for through-the-thickness deformations. The quality of the
present method in predicting static deformations and free vibrations of thin and thick laminated shells is compared and discussed
with other methods in some numerical examples.
2. Applying the Unified Formulation to SSDT
The Unified Formulation proposed by Carrera, also known as
CUF, is a powerful framework for the analysis of beams, plates
and shells. This formulation has been applied in several finite element analysis, either using the Principle of Virtual Displacements,
or by using the Reissner’s Mixed Variational theorem. The stiffness
matrix components, the external force terms or the inertia terms
can be obtained directly with this UF, irrespective of the shear
deformation theory being considered.
In this section, it is shown how to obtain the fundamental nuclei, which allows the derivation of the equations of motion and
boundary conditions according to CUF, in weak form for the finite
element analysis and in strong form for the present RBF
collocation.
3. A sinus shear deformation theory
The present sinus shear deformation theory involves the following expansion of displacements
pz
u3 ;
h
pz
w3
¼ w0 þ zw1 þ sin
h
u ¼ u0 þ zu1 þ sin
v ¼ v 0 þ zv 1 þ sin
pz
h
v 3;
w
ð1Þ
where u0, v0 and w0 are translations of a point at the middle-surface
of the plate, and u1, v1, u3, v3 denote rotations. This theory is an
expansion of early developments by Touratier [9–11], and Vidal
and Polit [12]. It considers a sinusoidal variation of all displacements u, v, w, allowing for through-the-thickness deformations.
Extension to shells becomes evident in the next sections.
the other two in-plane dimensions. Geometry and the reference
system are indicated in Fig. 1. The square of an infinitesimal linear
segment in the layer, the associated infinitesimal area and volume
are given by:
2
2
dV ¼
k
2
ð2Þ
Ha Hkb Hkz d
a db dz
where the metric coefficients are:
Hka ¼ Ak ð1 þ z=Rka Þ;
Hkb ¼ Bk ð1 þ z=Rkb Þ;
Hkz ¼ 1
ð3Þ
k
Rkb
k denotes the k-layer of the multilayered shell; Ra and
are the
principal radii of curvature along the coordinates a and b, respectively. Ak and Bk are the coefficients of the first fundamental form
of Xk (Ck is the Xk boundary). In this work, the attention has been
restricted to shells with constant radii of curvature (cylindrical,
spherical, toroidal geometries) for which Ak = Bk = 1.
Although one can use the UF for one-layer, isotropic shell, a
multi-layered shell with Nl layers is considered. The Principle of
Virtual Displacements (PVD) for the pure-mechanical case reads:
Nl Z
X
k¼1
Z n
Xk
Ak
dkpG
T
o
T
rkpC þ dknG rknC dXk dz ¼
Nl
X
dLke
ð4Þ
k¼1
where Xk and Ak are the integration domains in plane (a, b) and z
direction, respectively. Here, k indicates the layer and T the transpose of a vector, and dLke is the external work for the kth layer. G
means geometrical relations and C constitutive equations.
The steps to obtain the governing equations are:
Substitution of the geometrical relations (subscript G).
Substitution of the appropriate constitutive equations (subscript C).
Introduction of the Unified Formulation.
Stresses and strains are separated into in-plane and normal
components, denoted, respectively, by the subscripts p and n. The
mechanical strains in the kth layer can be related to the displacement field uk ¼ fuka ; ukb ; ukz g via the geometrical relations:
kpG ¼ ½kaa ; kbb ; kab T ¼ ðDkp þ Akp Þuk ; knG ¼ ½kaz ; kbz ; kzz T
¼ ðDknX þ Dknz Akn Þuk
ð5Þ
The explicit form of the introduced arrays follows:
2 @a
6
6
0
Dkp ¼ 6
6
4@
b
Shells are bi-dimensional structures in which one dimension (in
general the thickness in z direction) is negligible with respect to
2
dXk ¼ Hka Hkb da db
Hka
3.1. Governing equations and boundary conditions in the framework of
Unified Formulation
2
dsk ¼ Hka da2 þ Hkb db2 þ Hkz dz
Hkb
0
@b
Hkb
@a
Hka
0
3
7
7
0 7;
7
5
0
2
DknX
0 0
6
¼6
40 0
Fig. 1. Geometry and notations for a multilayered shell (doubly curved).
0 0
@a
Hka
3
7
@b 7
;
Hkb 5
0
2
@z
6
Dknz ¼ 4 0
0
0
@z
0
0
3
7
0 5;
@z
ð6Þ
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2
1
Hka Rka
0 0
6
k
6
Ap ¼ 4 0 0
3
2
7
1 7
;
Hkb Rkb 5
0 0
1
H k Rk
6 a a
Akn ¼ 6
4 0
0
0
0
1
Hkb Rkb
0
0
3
7
07
5
ð7Þ
0
The 3D constitutive equations are given as:
rkpC ¼ Ckpp kpG þ Ckpn knG
rknC ¼ Cknp kpG þ Cknn knG
ð8Þ
with
2
C k11
6 k
Ckpp ¼ 6
4 C 12
2
C k12
C k22
C k16
C k26
0
0
6
Cknp ¼ 4 0
C k13
0
C k23
C k16
3
2
7
C k26 7
5;
C k66
0
0 0 C k13
3
7
6
k 7
Ckpn ¼ 6
4 0 0 C 23 5;
2
3
0 0 C k36
C k55
6 k
Cknn ¼ 6
4 C 45
0
7
0 5;
k
C 36
C k45
C k44
0
0
3
ð9Þ
7
0 7
5
C k33
According to the Unified Formulation by Carrera, the three displacement components ua, ub and uz and their relative variations can be
modelled as:
ðua ; ub ; uz Þ ¼ F s ðuas ; ubs ; uzs Þ;
ðdua ; dub ; duz Þ
¼ F s ðduas ; dubs ; duzs Þ
ð10Þ
where the Fs are functions of the thickness coordinate z and s is a
sum index. Taylor expansions from first- up to fourth-order are employed: F0 = z0 = 1, F1 = z1 = z,. . ., FN = zN,. . ., F4 = z4 if an Equivalent
Single Layer (ESL) model is used. For ESL approach, one means that
the displacements are assumed for the whole laminate if a multilayer structure is considered.
Resorting to the displacement field in Eq. (10), the vectors
F t ¼ 1 z sin phz are chosen for the displacements u, v, w. Then,
all the terms of the equations of motion are obtained by integrating
through the thickness direction.
It is interesting to note that under this combination of the Unified Formulation and RBF collocation, the collocation code depends
only on the choice of Ft, in order to solve this type of problems. A
MATLAB code has been designed that, just by changing Ft, can analyse static deformations, and free vibrations for any type of C shear
deformation theory. An obvious advantage of the present methodology is that the tedious derivation of the equations of motion and
boundary conditions for a particular shear deformation theory is
no longer an issue, as this MATLAB code does all that work for us.
In Fig. 2, it is shown the assembling procedures of the stiffness
matrix on layer k for the ESL approach.
Substituting the geometrical relations, the constitutive equations and the Unified Formulation into the variational statement
PVD, for the kth layer, one has:
(Z
Nl
X
Z n
Xk
k¼1
T k
C pp ðDp þ Ap Þuk þ C kpn ðDnX þ Dnz An Þuk
ðDp þ Ap Þduk
Ak
T k
þ ðDnX þ Dnz An Þduk
C np ðDp þ Ap Þuk
) N
o
l
X
k
k
dLke
dXk dzk ¼
þC nn ðDnX þ Dnz An Þu
Fig. 2. Assembling procedure for ESL approach.
Z
X
T
ðDX Þdak ak dXk ¼ Xk
Z
Xk
þ
Z
ð11Þ
(Z
Nl
X
Xk
k¼1
Ck
where IX matrix is obtained applying the Gradient theorem:
Z n
h
dukT
ðDp þ Ap ÞT F s ðC kpp ðDp þ Ap ÞF s uks
s
Ak
i
þC kpn ðDnX þ Dnz An ÞF s uks Þ
h
ðDnX þ Dnz An ÞT F s ðC knp ðDp þ Ap ÞF s uks
þ dukT
s
io
o
þC knn ðDnX þ Dnz An ÞF s uks Þ dXk dzk
(Z Z
Nl
n
h
X
dukT
I Tp F s ðC kpp ðDp þ Ap ÞF s uks
þ
s
Ck
k¼1
Ak
i
h
þ dukT
I Tnp F s ðC knp ðDp Ap ÞF s uks
s
(Z
)
Nl
o X
io
k
þC knn ðDnX þ Dnz An ÞF s uks Þ dCk dzk ¼
dukT
F
p
s u
s
þC kpn ðDnX þ Dnz An ÞF s uks Þ
k¼1
ð14Þ
Xk
where Ikp and Iknp depend on the boundary geometry:
2 na
0
Ha
6
0
Ip ¼ 6
4
0
7
0 7;
5
0
nb
Hb
nb
Hb
3
na
Ha
2
Inp
0 0
6
¼ 40 0
na
Ha
nb
Hb
0 0
0
3
7
5
ð15Þ
^¼
n
na
nb
"
¼
cosðua Þ
#
ð16Þ
cosðub Þ
^ and the
where ua and ub are the angles between the normal n
direction a and b, respectively.
The governing equations for a multi-layered shell subjected to
mechanical loadings are:
T
dak ðDTX Þak dXk
T
dak ðIX Þak dCk
ð13Þ
The normal to the boundary of domain X is:
At this point, the formula of integration by parts is applied:
ni w ds
C
^ to the boundary along the
being ni the components of the normal n
direction i. After integration by parts and the substitution of CUF,
the governing equations and boundary conditions for the shell in
the mechanical case are obtained:
k¼1
Z
I
@w
dt ¼
@xi
ð12Þ
T
duks : Kkuuss uks ¼ Pkus
where the fundamental nucleus
ð17Þ
kss
Kuu
is obtained as:
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Z h
kss
K uu
¼
Ak
½Dp þ Ap T C kpp ½Dp þ Ap þ ½Dp þ Ap T C kpn ½DnX þ Dnz An þ ½DnX þ Dnz An T C knp ½Dp þ Ap þ½DnX þ Dnz An T
C knn ½DnX
þ Dnz An i
Ksuusk
23
1 k ss s
1 k ss s
J @ b C k22
J @ C k23 J kassz @ sb C k16
Rak
Rbk a=b b
1 kss s
1 k ss s
J @ C k26
J @ a C k36 J kbssz @ sa
Rak b=a a
Rbk
1 kss s
1 k ss s
þ C k45 J kbsz s @ sa J @ a þ C k44 J kasz s @ sb J a=b @ b
Rbk
Rbk
k
F s F s Ha Hkb dz
ð18Þ
and the corresponding Neumann-type boundary conditions on Ck
are:
ks
Pdkss uks ¼ Pdkss u
ð19Þ
Ksuusk
31
where
Z h
I Tp C kpp ½Dp þ Asp þ I Tp C kpn ½DnX þ Dnz Asn þ I Tnp C knp ½Dp þ Asp i
þI Tnp C knn ½DnX þ Dnz Asn F s F s Hka Hkb dz
ð20Þ
Pdkss ¼
¼ C k12
Ak
and Pkus are variationally consistent loads with applied pressure.
Ksuusk
3.2. Fundamental nuclei
32
The following integrals are introduced to perform the explicit
form of fundamental nuclei:
Z
H a Hb
J kss ; J kass ; J kbss ;J ka ss ; J kb ss ;J kasbs ¼
F s F s 1;Ha ; Hb ; ; ;Ha Hb dz
b
a
Hb H a
Ak
Z @F Ha Hb
s
ksz s ksz s ksz s ksz s ksz s ksz s
F s 1;Ha ;Hb ; ; ;Ha Hb dz
J ; J a ; J b ;J a ; J b ; J ab ¼
b
a
Hb Ha
Ak @z
Z
@F
H
s
a Hb
J kssz ; J kassz ; J kbssz ;J ka ssz ; J kb ssz ; J kasbsz ¼
Fs
1;Ha ;Hb ; ; ;Ha Hb dz
b
a
@z
Hb Ha
Ak
Z @F @F
s
s
J ksz sz ;J kasz sz ;J kbsz sz ;J ka sz sz ;J kb sz sz ;J kasbz sz ¼
b
a
Ak @z @z
Ha H b
ð21Þ
1; Ha ; Hb ; ; ; Ha Hb dz
H b Ha
Ksuusk
33
1 kss s
1 kss s
1 kss s
J @ þ C k12
J @ a þ C k13 J kbsz s @ sa þ C k16
J @b
Rak b=a a
Rbk
Rak
1 k ss s
1 k ss s
þ C k26
J a=b @ b þ C k36 J kasz s @ sb C k45 J kassz @ sb J @b
Rbk
Rak
1 k ss s
C k55 J kbssz @ sa J @
Rak b=a a
¼ C k11
1 kss s
1 kss s
1 kss s
J @ b þ C k22
J @ þ C k23 J kasz s @ sb þ C k16
J @
Rak
Rbk a=b b
Rak b=a a
1 k ss s
1 kss s
þ C k26
J @ a þ C k36 J bksz s @ sa C k45 J kbssz @ sa J @a
Rbk
Rbk
1
J k ss @ s
C k44 J kassz @ sb Rbk a=b b
¼ C k12
¼ C k11
1
2
J kb=sas þ C k22
1
J kas=bs
R2bk
þ C k33 J kasbz sz þ 2C k12
1
Rak
Rak
1 k ss
1 k sz s
J þ C k13
J b þ J kbssz þ C k23
Rbk
Rak
1 k sz s
k ss s s
J a þ J kassz C k44 J kas=bs @ sb @ sb C k55 J b=
a@a@a
Rbk
C k45 J kss @ sa @ sb C k45 J kss @ sa @ sb
ð22Þ
The fundamental nuclei Kkuuss is reported for doubly curved shells (radii of curvature in both a and b directions (see Fig. 1)):
The application of boundary conditions makes use of the fundamental nuclei Pd in the form:
Ksuusk
ssk k k ss s
k k ss s
k k ss s
k ss s
Puu 11 ¼ na C k11 J b=
a @ a þ nb C 66 J a=b @ b þ nb C 16 J @ a þ na C 16 J @ b
ssk k k ss s
k kss s
k kss s
k ss s
Puu 12 ¼ na C k16 J b=
a @ a þ nb C 26 J a=b @ b þ na C 12 J @ b þ nb C 66 J @ a
ssk 1 k kss
1 k k ss
1 k kss
Puu 13 ¼ na
C J þ na
C J þ na C k13 J kbssz þ nb
C J
Rak 11 b=a
Rbk 12
Rak 16
1 k k ss
þ nb
C J þ nb C k36 J kassz
Rbk 26 a=b
ssk k k ss s
k k ss s
k k ss s
k ss s
Puu 21 ¼ na C k16 J b=
a @ a þ nb C 26 J a=b @ b þ nb C 12 J @ a þ na C 66 J @ b
ssk k k ss s
k k ss s
k k ss s
Puu 22 ¼ na C 66 J b=a @ a þ nb C 22 J a=b @ b þ nb C 26 J @ a þ na C k26 Jkss @ sb
ssk 1 k kss
1 k k ss
1 k kss
Puu 23 ¼ na
C J þ na
C J þ na C k36 J kbssz þ nb
C J
Rak 16 b=a
Rbk 26
Rak 12
1 k k ss
þ nb
C J þ nb C k23 J kassz
Rbk 22 a=b
ssk 1 k k ss
1 k kss
Puu 31 ¼ na
C J þ na C k55 J kbssz nb
C J þ nb C k45 J kassz
Rak 55 b=a
Rak 45
ssk 1 k k ss
1 k k ss
Puu 32 ¼ na
C J þ na C k45 J kbssz nb
C J þ nb C k44 J kassz
Rbk 45
Rbk 44 a=b
ssk k k ss s
k k ss s
k k ss s
k ss s
Puu 33 ¼ na C k55 J b=
a @ a þ nb C 44 J a=b @ b þ nb C 45 J @ a þ na C 45 J @ b ð23Þ
11
k k ss s s
k k ss s s
k kss s s
k ss s s
¼ C k11 J b=
a @ a @ a C 16 J @ a @ b C 16 J @ a @ b C 66 J a=b @ b @ b
!
1 ksz s
1 kssz
1 k ss
þ C k55 J kasbz sz J
J
þ 2 J b=
a
Rak b
Rak b
Ra
k
Ksuusk
12
Ksuusk
13
Ksuusk
21
Ksuusk
22
k k ss s s
k kss s s
k ss s s
¼ C k12 J kss @ sa @ sb C k16 J b=
a @ a @ a C 26 J a=b @ b @ b C 66 J @ a @ b
1 k sz s
1 k ssz
1 1 kss
þ C k45 J kasbz sz Ja Jb þ
J
Rbk
Rak
Rak Rbk
1 k ss s
1 k ss s
1 k ss s
J @ C k12
J @ a C k13 J kbssz @ sa C k16
J @b
Rak b=a a
Rbk
Rak
1 k ss s
1 k ss s
C k26
Ja=b @ b C k36 J kassz @ sb þ C k45 J kasz s @ sb J @b
Rbk
Rak
1 k ss s
þ C k55 J bksz s @ sa J @
Rak b=a a
¼ C k11
k k ss s s
k kss s s
k ss s s
¼ C k12 J kss @ sa @ sb C k16 J b=
a @ a @ a C 26 J a=b @ b @ b C 66 J @ a @ b
1 k ssz
1 k sz s
1 1 kss
þ C k45 J kasbz sz J
J
þ
J
Rbk a
Rak b
Rak Rbk
¼ C k22 J kas=bs @ sb @ sb C k26 J kss @ sa @ sb C k26 J kss @ sa @ sb C k66 J kb=sas @ sa @ sa
!
1 k sz s
1 k ssz
1 kss
k
ksz sz
þ C 44 J ab J
J
þ 2 J a=b
Rbk a
Rbk a
Rb
k
One can note that all the equations written for the shell degenerate
in those for the plate when R1 ¼ R1 ¼ 0. In practice, the radii of curak
bk
vature are set to 109.
3.3. Dynamic governing equations
The PVD for the dynamic case is expressed as:
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Nl Z
X
Z n
Xk
k¼1
¼
Ak
Nl Z
X
k¼1
dkpG
Z
Xk
T
T
o
4.2. The eigenproblem
rkpC þ dknG rknC dXk dz
k
kT € k
q du u dXk dz þ
Ak
Nl
X
dLke
ð24Þ
k¼1
where qk is the mass density of the kth layer and double dots denote acceleration.
By substituting the geometrical relations, the constitutive equations and the Unified Formulation, one obtains the following governing equations:
The eigenproblem looks for eigenvalues (k) and eigenvectors (u)
that satisfy
Lu þ ku ¼ 0 in X
ð30Þ
LB u ¼ 0 on @ X
ð31Þ
As in the static problem, the eigenproblem defined in (30) and (31)
is replaced by a finite-dimensional eigenvalue problem, based on
RBF approximations.
4.3. Radial basis functions approximations
T
k ss k
€ ks þ Pkus
duks : Kuu
us ¼ Mkss u
ð25Þ
The radial basis function (/) approximation of a function (u) is
given by
In the case of free vibrations one has:
T
k ss k
€ ks
duks : Kuu
us ¼ Mkss u
ð26Þ
~ ðxÞ ¼
u
N
X
ai /ðkx yi k2 Þ; x 2 Rn
ð32Þ
i¼1
where Mkss is the fundamental nucleus for the inertial term. The explicit form of that is:
where yi, i = 1, . . . , N is a finite set of distinct points (centers) in Rn .
The most common RBFs are
/ðrÞ ¼ r 3
k ss
M11
¼ qk J kasbs
Cubic :
k ss
M12
¼0
Thin plate splines :
Wendland functions :
k ss
M13
¼0
Gaussian :
k ss
¼0
M21
k ss
M22
¼ qk J kasbs
ð27Þ
k ss
¼0
M31
¼0
k ss
M33
¼ qk J kasbs
/ðrÞ ¼ eðcrÞ
Multiquadrics :
/ðrÞ ¼
Inverse multiquadrics :
k ss
M23
¼0
k ss
M32
/ðrÞ ¼ r2 logðrÞ
where the meaning of the integral Jkasbs has been illustrated in Eq.
(21). The geometrical and mechanical boundary conditions are the
same of the static case. Because the static case is only considered,
the mass terms will be neglected.
/ðrÞ ¼ ð1 rÞm
þ pðrÞ
2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c2 þ r 2
/ðrÞ ¼ ðc2 þ r 2 Þ1=2
where the Euclidian distance r is real and non-negative and c is a
positive shape parameter. Hardy [45] introduced multiquadrics in
the analysis of scattered geographical data. In the 1990s Kansa
[26] used multiquadrics for the solution of partial differential equations. Considering N distinct interpolations, and knowing u(xj),
j = 1, 2, . . . , N, one finds ai by the solution of a N N linear system
Aa ¼ u
where
A = [/(kx yik2)]NN,
u = [u(x1), u(x2), . . . , u(xN)]T.
ð33Þ
a = [a1, a2, . . . , aN]
T
and
4. The radial basis function method
4.4. Solution of the static problem
4.1. The static problem
The solution of a static problem by radial basis functions considers NI nodes in the domain and NB nodes on the boundary, with
a total number of nodes N = NI + NB. One can denote the sampling
points by xi 2 X, i = 1, . . . , NI and xi 2 oX, i = NI + 1, . . . , N. At the
points in the domain, the following system of equations is solved
Radial basis functions (RBF) approximations are mesh-free
numerical schemes that can exploit accurate representations of
the boundary, are easy to implement and can be spectrally accurate. In this section the formulation of a global unsymmetrical collocation RBF-based method to compute elliptic operators is
presented.
Consider a linear elliptic partial differential operator L and a
bounded region X in Rn with some boundary oX. In the static
problems, the displacements (u) are computed from the global system of equations
Lu ¼ f in X
ð28Þ
LB u ¼ g on @ X
ð29Þ
where L; LB are linear operators in the domain and on the boundary, respectively. The right-hand side of (28) and (29) represent the
external forces applied on the plate or shell and the boundary conditions applied along the perimeter of the plate or shell, respectively. The PDE (Partial Differential Equation) problem defined in
(28) and (29) will be replaced by a finite problem, defined by an
algebraic system of equations, after the radial basis expansions.
N
X
ai L/ðkx yi k2 Þ ¼ fðxj Þ; j ¼ 1; 2; . . . ; NI
ð34Þ
i¼1
or
LI a ¼ F
ð35Þ
where
LI ¼ ½L/ðkx yi k2 ÞNI N
ð36Þ
At the points on the boundary, the following boundary conditions
are imposed
N
X
ai LB /ðkx yi k2 Þ ¼ gðxj Þ; j ¼ NI þ 1; . . . ; N
ð37Þ
i¼1
or
Ba ¼ G
ð38Þ
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tion w = 0, on a simply supported or clamped edge. The conditions
are enforced by interpolating as
where
B ¼ LB / kxNI þ1 yj k2 N
B N
w¼0!
LI
#
a¼
B
ð39Þ
G
4.5. Solution of the eigenproblem
ai L/ðkx yi k2 Þ ¼ ku~ ðxj Þ; j ¼ 1; 2; . . . ; NI
ð40Þ
i¼1
or
~I
LI a ¼ ku
v 0 ¼ V 0 ðx; yÞe
u1 ¼ U 1 ðx; yÞeixt ;
v 1 ¼ V 1 ðx; yÞe
;
w0 ¼ W 0 ðx; yÞeixt ;
ixt
;
w1 ¼ W 1 ðx; yÞeixt ;
u3 ¼ U 3 ðx; yÞeixt
ð50Þ
v 3 ¼ V 3 ðx; yÞeixt
ð51Þ
w3 ¼ W 3 ðx; yÞeixt
ð52Þ
where x is the frequency of natural vibration. Substituting the harmonic expansion into Eq. (45) in terms of the amplitudes U0, U1, U3,
V0, V1, V3, W0, W1, W3, one can obtain the natural frequencies and
vibration modes for the plate or shell problem, by solving the
eigenproblem
ð42Þ
where L collects all stiffness terms and G collects all terms related
to the inertial terms. In (53) X are the modes of vibration associated
with the natural frequencies defined as x.
ð43Þ
i¼1
ð53Þ
5. Numerical examples
All numerical examples consider a Chebyshev grid (see Fig. 3)
and a Wendland function, defined as
/ðrÞ ¼ ð1 crÞ8þ 32ðcrÞ3 þ 25ðcrÞ2 þ 8cr þ 1
or
Ba ¼ 0
ð44Þ
Eqs. (41) and (44) can now be solved as a generalized eigenvalue
problem
"
ix t
L x2 G X ¼ 0
At the points on the boundary, the imposed boundary conditions
are
ai LB /ðkx yi k2 Þ ¼ 0; j ¼ NI þ 1; . . . ; N
For free vibration problems, the external forces are set to zero,
and an harmonic solution is assumed for the displacements u0,
u1, v0, v1, . . .
ð41Þ
where
LI ¼ ½L/ðkx yi k2 ÞNI N
4.7. Free vibrations problems
u0 ¼ U 0 ðx; yÞeixt ;
Consider NI nodes in the interior of the domain and NB nodes on
the boundary, with N = NI + NB. The interpolation points are denoted by xi 2 X, i = 1, . . . , NI and xi 2 oX, i = NI + 1, . . . , N. At the
points in the domain, the following eigenproblem is defined
N
X
ð49Þ
Other boundary conditions are interpolated in a similar way.
F
By inverting the system (39), one obtains the vector a. Then, the
solution u is calculated using the interpolation Eq. (32).
N
X
aW
j /j ¼ 0
j¼1
Therefore, one can write a finite-dimensional static problem as
"
N
X
LI
#
B
"
a¼k
AI
#
0
ð54Þ
where the shape parameter (c) was obtained by an optimization
procedure, as detailed in Ferreira and Fasshauer [46].
5.1. Spherical shell in bending
a
ð45Þ
where
A laminated composite spherical shell is here considered, of
side a and thickness h, composed of layers oriented at [0°/90°/0°]
AI ¼ / ðkxNI yj k2 Þ N N
Chebyshev grid
I
1
4.6. Discretization of the equations of motion and boundary conditions
0.8
0.6
The radial basis collocation method follows a simple implementation procedure. Taking Eq. (39), one computes
a¼
#1 F
L
G
B
I
ð46Þ
~ , by using (32). If
This a vector is then used to obtain solution u
~ are needed, such derivatives are computed as
derivatives of u
0.4
y−coordinates
"
0.2
0
−0.2
−0.4
N
~ X
@/
@u
¼
aj j
@x
@x
j¼1
ð47Þ
N
~ X
@2/
@2u
¼
aj 2j ; etc:
2
@x
@x
j¼1
ð48Þ
−0.6
−0.8
In the present collocation approach, one needs to impose essential
and natural boundary conditions. Consider, for example, the condi-
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
x−coordinates
Fig. 3. A sketch of a Chebyshev grid for 13 13 points.
0.8
1
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A.J.M. Ferreira et al. / Composites: Part B 42 (2011) 1276–1284
Table 1
3
2
¼ w 10P Ea24h variation with various number of grid points per unit length, N for different R/a ratios, for R1 = R2.
Non-dimensional central deflection, w
0
a/h
[0°/90°/0°]
[0°/90°/90°/0°]
Method
R/a
5
10
20
50
100
109
10
10
10
10
10
Present (13 13)
Present (17 17)
Present (21 21)
HSDT [47]
FSDT [47]
6.6874
6.6879
6.6880
6.7688
6.4253
6.9044
6.9047
6.9048
7.0325
6.6247
6.9615
6.9618
6.9618
7.1016
6.6756
6.9781
6.9784
6.9784
7.1212
6.6902
6.9806
6.9809
6.9809
7.1240
6.6923
6.9816
6.9819
6.9820
7.125
6.6939
100
100
100
100
100
Present (13 13)
Present (17 17)
Present (21 21)
HSDT [47]
FSDT [47]
1.0244
1.0249
1.0250
1.0321
1.0337
2.3651
2.3661
2.3662
2.4099
2.4109
3.5151
3.5165
3.5167
3.617
3.6150
4.0692
4.0707
4.0709
4.2071
4.2027
4.1629
4.1644
4.1646
4.3074
4.3026
4.1951
4.1966
4.1966
4.3420
4.3370
10
10
10
10
10
Present (13 13)
Present (17 17)
Present (21 21)
HSDT [47]
FSDT [47]
6.7199
6.7204
6.7205
6.7865
6.3623
6.9418
6.9423
6.9423
7.0536
6.5595
7.0004
7.0007
7.0008
7.1237
6.6099
7.0174
7.0178
7.0178
7.1436
6.6244
7.0201
7.0204
7.0204
7.1464
6.6264
7.0211
7.0214
7.0215
7.1474
6.6280
100
100
100
100
100
Present (13 13)
Present (17 17)
Present (21 21)
HSDT [47]
FSDT [47]
1.0190
1.0195
1.0195
1.0264
1.0279
2.3581
2.3591
2.3592
2.4024
2.4030
3.5119
3.5132
3.5134
3.6133
3.6104
4.0694
4.0708
4.0711
4.2071
4.2015
4.1638
4.1653
4.1655
4.3082
4.3021
4.1962
4.1978
4.1980
4.3430
4.3368
Table 2
pffiffiffiffiffiffiffiffiffiffiffi
¼ x ah2 q=E2 , laminate ([0°/90°/90°/0°]).
Nondimensionalized fundamental frequencies of cross-ply laminated spherical shells, x
a/h
Method
R/a
5
10
20
50
100
109
10
Present (13 13)
Present (17 17)
Present (21 21)
HSDT [47]
12.0999
12.0995
12.0994
12.040
11.9378
11.9375
11.9375
11.840
11.8967
11.8964
11.8964
11.790
11.8851
11.8849
11.8849
11.780
11.8835
11.8832
11.8832
11.780
11.8829
11.8827
11.8827
11.780
100
Present (13 13)
Present (17 17)
Present (21 21)
HSDT [47]
31.2175
31.2076
31.2063
31.100
20.5753
20.5690
20.5683
20.380
16.8713
16.8663
16.8658
16.630
15.6760
15.6714
15.6711
15.420
15.4977
15.4931
15.4929
15.230
15.4378
15.4333
15.4331
15.170
Table 3
pffiffiffiffiffiffiffiffiffiffiffi
¼ x ah2 q=E2 , laminate ([0°/90°/0°]).
Nondimensionalized fundamental frequencies of cross-ply laminated spherical shells, x
a/h
Method
R/a
5
10
20
50
100
109
10
Present (13 13)
Present (17 17)
Present (21 21)
HSDT [47]
12.1258
12.1254
12.1253
12.060
11.9661
11.9658
11.9658
11.860
11.9256
11.9253
11.9253
11.810
11.9142
11.9140
11.9140
11.790
11.9126
11.9123
11.9123
11.790
11.9120
11.9112
11.9112
11.790
100
Present (13 13)
Present (17 17)
Present (21 21)
HSDT [47]
31.1360
31.1262
31.1249
31.020
20.5441
20.5388
20.5381
20.350
16.8634
16.8584
16.8579
16.620
15.6764
15.6718
15.6714
15.420
15.4993
15.4948
15.4944
15.240
15.4398
15.4353
15.4349
15.170
and [0°/90°/90°/0°]. The shell is subjected to a sinusoidal vertical
pressure of the form
pz ¼ P sin
px
a
sin
py
¼
a
with the origin of the coordinate system located at the lower left
corner on the midplane and P the maximum load (at center of shell).
The orthotropic material properties for each layer are given by
E1 ¼ 25:0E2 ;
G12 ¼ G13 ¼ 0:5E2 ;
3
¼
w
G23 ¼ 0:2E2 ;
m12 ¼ 0:25
The in-plane displacements, the transverse displacements, the normal stresses and the in-plane and transverse shear stresses are presented in normalized form as
103 wða=2;a=2;0Þ h E2
Pa4
ryyða=2;a=2;h=4Þ h2
Pa
2
;
;
r xx ¼
sxz ¼
rxxða=2;a=2;h=2Þ h2
Pa2
sxzð0;a=2;0Þ h
Pa
;
;
sxy ¼
r yy
sxyð0;0;h=2Þ h2
Pa2
The shell is simply supported on all edges.
In Table 1, the static deflections for the present shell model are
compared with results of Reddy shell formulation using first-order
and third-order shear deformation theories [47]. Nodal grids with
13 13, 17 17, and 21 21 points are considered. Various values of R/a and two values of a/h (10 and 100) are taken for the analysis. Results are in good agreement for various a/h ratios with the
higher-order results of Reddy [47].
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Table 4
pffiffiffiffiffiffiffiffiffiffiffi
¼ x ah2 q=E2 .
Nondimensionalized fundamental frequencies of cross-ply cylindrical shells, x
R/a
Method
[0/90/0]
[0/90/90/0]
a/h = 100
a/h = 10
a/h = 100
5
Present (13 13)
Present (17 17)
Present (21 21)
FSDT [47]
HSDT [47]
20.4988
20.4871
20.4856
20.332
20.330
11.9234
11.9230
11.9230
12.207
11.850
20.5310
20.5214
20.5202
20.361
20.360
11.9010
11.9007
11.9007
12.267
11.830
10
Present (13 13)
Present (17 17)
Present (21 21)
FSDT [47]
HSDT [47]
16.8448
16.8448
16.8441
16.625
16.620
11.9149
11.9146
11.9146
12.173
11.800
16.8598
16.8538
16.8531
16.634
16.630
11.8874
11.8872
11.8872
12.236
11.790
20
Present (13 13)
Present (17 17)
Present (21 21)
FSDT [47]
HSDT [47]
15.8048
15.7998
15.7993
15.556
15.55
11.9127
11.9125
11.9125
12.166
11.79
15.8055
15.8006
15.8001
15.559
15.55
11.8840
11.8838
11.8838
12.230
11.78
50
Present (13 13)
Present (17 17)
Present (21 21)
FSDT [47]
HSDT [47]
15.4988
15.4942
15.4938
15.244
15.24
11.9121
11.9119
11.9119
12.163
11.79
15.4972
15.4926
15.4922
15.245
15.23
11.8831
11.8829
11.8829
12.228
11.78
100
Present (13 13)
Present (17 17)
Present (21 21)
FSDT [47]
HSDT [47]
15.4546
15.4501
15.4497
15.198
15.19
11.9120
11.9118
11.9118
12.163
11.79
15.4527
15.4481
15.4477
15.199
15.19
11.8830
11.8827
11.8827
12.227
11.78
Plate
Present (13 13)
Present (17 17)
Present (21 21)
FSDT [47]
HSDT [47]
15.4398
15.4353
15.4349
15.183
15.170
11.9120
11.9118
11.9118
12.162
11.790
15.4378
15.4333
15.4328
15.184
15.170
11.8829
11.8827
11.9927
12226
11.780
a/h = 10
5.2. Free vibration of spherical and cylindrical laminated shells
eig = 33.239362338432
eig = 20.575318444535
1
y−coordinate
y−coordinate
1
0.5
0
−0.5
−1
−1
0
0.5
0
−0.5
−1
−1
1
eig = 56.520391579289
1
eig = 60.338036781464
1
y−coordinate
1
0.5
0
−0.5
−1
−1
0
x−coordinate
x−coordinate
y−coordinate
Nodal grids with 13 13, 17 17, and 21 21 points are considered. In Tables 2 and 3 the nondimensionalized natural frequencies from the present SSDT theory for various cross-ply spherical
shells are compared with analytical solutions by Reddy and Liu
[47], who considered both the First-order Shear Deformation Theory (FSDT) and the High-order Shear Deformation Theory (HSDT).
The first-order theory overpredicts the fundamental natural frequencies of symmetric thick shells and symmetric shallow thin
shells. The present radial basis function method is compared with
analytical results by Reddy [47] and shows excellent agreement.
Table 4 contain nondimensionalized natural frequencies obtained using the present SSDT theory for cross-ply cylindrical
shells with lamination schemes [0/90/0], [0/90/90/0]. Present results are compared with analytical solutions by Reddy and Liu
[47] who considered both the first-order (FSDT) and the third-order (HSDT) theories. The present radial basis function method is
compared with analytical results by Reddy [47] and shows excellent agreement.
In Fig. 4, the first four p
vibrational
modes of cross-ply laminated
ffiffiffiffiffiffiffiffiffiffiffi
¼ x ah2 q=E2 , are illustrated for a laminate ([0°/
spherical shells, x
90°/90°/0°]), using a grid of 13 13 points, for a/h = 100, R/a = 10.
The modes of vibration are quite stable.
0
x−coordinate
1
0.5
0
−0.5
−1
−1
0
1
x−coordinate
6. Concluding remarks
Fig. 4. First
four
vibrational modes of cross-ply laminated spherical shells,
pffiffiffiffiffiffiffiffiffiffi
ffi
¼ x ah2 q=E2 , laminate ([0°/90°/90°/0°]) grid 13 13 points, a/h = 100,R/a = 10.
x
In this paper a sinusoidal shear deformation theory was implemented for the first time for laminated orthotropic elastic shells
through a multiquadrics discretization of equations of motion
and boundary conditions. The multiquadric radial basis function
method for the solution of shell bending and free vibration prob-
lems was presented. Results for static deformations and natural
frequencies were obtained and compared with other sources. This
meshless approach demonstrated that is very successful in the static deformations and free vibration analysis of laminated compos-
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A.J.M. Ferreira et al. / Composites: Part B 42 (2011) 1276–1284
ite shells. Advantages of radial basis functions are absence of mesh,
ease of discretization of boundary conditions and equations of
equilibrium or motion and very easy coding. The static displacements and the natural frequencies obtained from present method
are shown to be in excellent agreement with analytical solutions.
Acknowledgements
The support of Ministério da Ciência Tecnologia e do Ensino
superior and Fundo Social Europeu (MCTES and FSE) under programs POPH-QREN are gratefully acknowledged.
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